Maximum value of `f(x)=-(x-1)^(2)+2` is ………. .

Answer

To find the maximum value of the function \( f(x) = -(x - 1)^2 + 2 \), we can follow these steps:

### Step 1: Identify the function
We have the function:
\[ f(x) = -(x - 1)^2 + 2 \]

### Step 2: Differentiate the function
To find the critical points, we need to differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[-(x - 1)^2 + 2] \]
Using the chain rule, we differentiate:
\[ f'(x) = -2(x - 1) \cdot \frac{d}{dx}(x - 1) = -2(x - 1) \cdot 1 = -2(x - 1) \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ -2(x - 1) = 0 \]
Solving for \( x \):
\[ x - 1 = 0 \implies x = 1 \]

### Step 4: Determine if it is a maximum or minimum
To determine if this critical point is a maximum or minimum, we can use the second derivative test. We differentiate \( f'(x) \):
\[ f''(x) = \frac{d}{dx}[-2(x - 1)] = -2 \]
Since \( f''(x) = -2 < 0 \), this indicates that the function is concave down at \( x = 1 \), confirming that it is a maximum.

### Step 5: Calculate the maximum value
Now, we find the maximum value by substituting \( x = 1 \) back into the original function:
\[ f(1) = - (1 - 1)^2 + 2 = -0 + 2 = 2 \]

### Conclusion
The maximum value of the function \( f(x) = -(x - 1)^2 + 2 \) is:
\[ \boxed{2} \]

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